Concave utility functions are widely used in applied economics and in economic theory. A complete characterization of preferences representable by such utility functions has been known for some time, provided that the preferences are defined on infinite convex sets. The real world is finite. This motivates recent work by Richter and Wong concerning convex preferences defined on finite sets. We provide geometric constructions and clarify the relation with older algorithms in demand theory. We also show that certain natural objects (such as least concave utility functions) do not exist in general in the finite context.